Problem 70 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$12+2\left(\frac{12-2}{7-2}\right)-3\left(\frac{12-9}{17-14}\right)$$

Step-by-Step Solution

Answer

The simplified expression is 13.

Step 1: Simplify Fractions

First, simplify the fractions inside the parentheses \(\left(\frac{12-2}{7-2}\right)\) and \(\left(\frac{12-9}{17-14}\right)\). For the first fraction: \(\frac{12-2}{7-2} = \frac{10}{5} = 2\). For the second fraction: \(\frac{12-9}{17-14} = \frac{3}{3} = 1\). Replace the fractions in the original expression with these simplified values.

Step 2: Substitute Simplified Values into the Expression

Substitute the simplified values back into the original expression. This gives us: \(12 + 2 \times 2 - 3 \times 1\).

Step 3: Perform Multiplication

Perform the multiplication in the expression. Calculate \(2 \times 2 = 4\) and \(3 \times 1 = 3\). This transforms the expression into \(12 + 4 - 3\).

Step 4: Simplify the Expression

Perform the addition and subtraction operations to simplify the expression. Add first: \(12 + 4 = 16\). Then subtract: \(16 - 3 = 13\).

Key Concepts

Order of Operations

The order of operations is a fundamental principle in mathematics that ensures consistency in solving arithmetic problems. It is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding and applying the correct order of operations is crucial to simplifying numerical expressions correctly.
When working with complex arithmetic expressions, remember to:

• First, evaluate expressions inside parentheses. This also includes simplifying any fractions or other operations inside the parentheses before moving on to the next step.
• Next, handle any exponents.
• After that, perform multidigit multiplication or division, from left to right.
• Finally, resolve addition and subtraction from left to right.

In the provided exercise, the operations are mainly inside the parentheses and involve fractions, making it essential to simplify within the parentheses first before continuing with multiplication.

Fractions Simplification

Simplifying fractions is an important step in handling numerical expressions, especially when they appear inside parentheses. A fraction is simplified when both the numerator (top number) and the denominator (bottom number) have no common factors other than 1. Simplifying fractions makes equations and expressions easier to work with.
Steps to simplify fractions:

• Subtract or add any numbers in the numerator or denominator first (as per the expression requirements).
• Look for common factors in both the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common divisor (GCD).

In the example given, the expressions inside the parentheses are simplified first. For example, \((12 - 2)/(7 - 2) = 10/5 = 2\) and \((12 - 9)/(17 - 14) = 3/3 = 1\). These steps simplify the fractions, leading to further simplification in subsequent steps.

Substitution Method

The substitution method is useful in mathematics to replace variables or complex parts of an expression with simpler equivalents. This makes calculations straightforward and minimizes errors. In numerical expressions, once you simplify components (like fractions), you substitute them back into the main expression to continue simplifying.
Here's how it’s done:

• Simplify the parts of the expression independently, if needed (such as fractions or components inside parentheses).
• Substitute these simplified parts back into the original equation.
• Ensure the new expression is correct and ready for further simplification.

In the problem we're solving, after simplifying the fractions to 2 and 1, these values are substituted back into the original expression, resulting in \(12 + 2 \times 2 - 3 \times 1\). This allows us to proceed smoothly to multiplication next.

Arithmetic Operations

Arithmetic operations involve basic mathematical functions such as addition, subtraction, multiplication, and division. These operations are the building blocks of more complex calculations, and understanding how to work them out accurately is vital for solving any numerical expressions.
Here's a quick guide:

• Multiplication and division approach each other equally and should be performed first, following any necessary parentheses.
• Addition and subtraction come next, processed from left to right.

In the exercise, after substitution and simplifying operations within parentheses, the next steps are to multiply (\(2 \times 2 = 4\)) and (\(3 \times 1 = 3\)), followed by addition and subtraction (\(12 + 4 - 3 = 13\)). Proper execution ensures the accuracy of the simplified expression.